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15    WEAK TOPOLOGIES       75

v is a continuous linear mapping of F into a locally convex space G, then

(12.15.6)                                  '(z; o w) = 'w o '*;.

The weak topology on the dual of a locally convex space is not metrizable
in general (Problem 2). However, there is the following result:

(12.15.7)    Let E be a separable metrizable locally convex space. IfH is any
equicontinuous (7.5) subset of the dual E' o/E, then the weak closure ofH in E7
is a compact metrizable space in the weak topology.

We shall first prove the following lemma:

(1 2.1 5.7.1 ) Let E be a metrizable vector space and F a normed space. In order
that a set H of linear mappings ofE into F should be equicontinuous, it is neces-
sary and sufficient that there should exist a neighborhood V of in E and a real
number c> 0 such that \\u(x)\\  cfor all x e V and all ueH.

(If E is normed, this condition is also equivalent to sup \\u\\ < H-oo

weH

by (5.7.1).)

The condition expresses that H is equicontinuous at the point 0. If this
is the case, then for each x0 e E and each x e x0 4- eV, we have

\\U(X)-U(XO)\\ = \\U(X-XO)\\BC
for all u e H, and therefore H is equicontinuous at x0 .

In proving (12.15,7), we may first of all restrict ourselves to the case where
H is weakly closed in E'. For if fl is the weak closure of H in E', the continuity
of the mapping X'H><JC, x") on E' implies that if |<X *')| ^ c for all xe Vand
all xf e H, then also |<x, x')\  c for all jc e V and all x' e H (3.15.4), whence
the result by (12.15.7.1).

Now let (an) be a sequence which is dense in E. We shall show that the
weak topology on H is defined by the pseudo-distances

Clearly the topology defined by these pseudo-distances is coarser than the
weak topology; hence it is enough to show that it is also finer. By hypothesis,
every neighborhood of a point x'0 e H in the weak topology contains a subset
W of H consisting of elements x' e H satisfying & finite number of inequalities

(12.15.7.2)              |<jt,, x' - *i>| < r       (lim)k topology.